Algebraic Fractions Complete Guide with Concepts and Examples

Algebraic fractions, such as having a variable in the numerator or denominator. Because division by 0 is impossible, variables in the denominator are subject to specific constraints. The denominator can never be zero. Therefore, algebraic fractions follow the same rules as arithmetic fractions. There are two kinds of rational algebraic fractions: Proper algebraic fractions and improper algebraic fractions.

Proper algebraic fractions are those that have a numerator with a lower degree than the denominator.

Improper algebraic fractions are those in which the numerator has a greater degree than the denominator.

Simplifying Algebraic Fractions

Sometimes the top and bottom of a fraction can be divided by the same number. This is called cancelling down. It is also called simplifying the fraction. Fractions often have to be written in their simplest terms. This means they have to be cancelled until they cannot be cancelled any more.

To do this, look for fractions where the numerator (top number) and the denominator (bottom number) are both multiples of the same times table. This tells you their common factor, which you use to divide the top and bottom number, in order to simplify or cancel down the fraction as required.

Calculate $\dfrac{2}{3}-\dfrac{y}{18}$

Ans: $=\dfrac{2 \times 18}{54}-\dfrac{3 y}{54}$

$=\dfrac{36}{54}-\dfrac{3 y}{54}$

$=\dfrac{36-3 y}{54}$

$=\dfrac{3(12-y)}{54}$

Take out a common factor of 3 on the numerator, then you notice that you can simplify by dividing top and bottom by 3 .

$=\dfrac{12-y}{18}$

Adding and Subtracting Algebraic Fractions

When adding and subtracting fractions, we must ensure that we have the same denominator.

For adding and subtracting fractions:

Step 1: Multiply the two terms on the bottom to get the same denominator.

Step 2: Multiply the top number on the first fraction by the bottom number of the second fraction to get the new top number of the first fraction.

Step 3: Multiply the top number on the second fraction by the bottom number of the first fraction to get the new top number of the second fraction.

Step 4: Now add/subtract the top numbers and keep the bottom number, so there is one fraction.

Step 5: Simplify the fraction if required.

Example

Calculate: $\dfrac{2}{5}+\dfrac{3}{7}$

$=\dfrac{2 \times 7}{35}+\dfrac{3 \times 5}{35}$

$=\dfrac{14}{35}+\dfrac{15}{35}=\dfrac{29}{35}$

Multiplying Algebraic Fractions

To multiply algebraic fractions, factorise the numerators and denominators. Then cancel the factors common to the numerator and denominator before applying multiplication to obtain the answer.

Simplify $\dfrac{4}{a+a^2} \times \dfrac{a^3-a}{a b}$

Solution:

$= frac{4}{a+a^2} \times \dfrac{a^3-a}{a b}$ (Factorise the numerator and denominator)

$= \dfrac{4}{a(a+1)} \times \dfrac{a\left(a^2-1\right)}{a b}$ (Difference of two squares)

$= \dfrac{4}{a(a+1)} \times \dfrac{a(a+1)(a-1)}{a b}$ (Cancel the common factors)

$= \dfrac{4(a-1)}{a b}$

Dividing Algebraic Fractions

To divide algebraic fractions, invert the second fraction and multiply it by the first fraction. Factorise the numerators and denominators. Then cancel the factors common to both the numerator and denominator before applying multiplication to obtain the answer.

For example, $\dfrac{a}{16} \div \dfrac{a}{12}$

Ans: $\dfrac{a}{16} \div \dfrac{a}{12}$

$=\dfrac{a}{16} \times \dfrac{12}{a}$

$=\dfrac{3}{4}$

Fraction Numbers Arithmetic Properties

1. Fractions Addition with Equal Denominators

$\dfrac{A}{B}+\dfrac{C}{B}=\dfrac{A+C}{B}$

2. Fractions Addition with Different Denominators

$\dfrac{\mathrm{A}}{\mathrm{B}}+\dfrac{\mathrm{C}}{\mathrm{D}}=\dfrac{\mathrm{AD}+\mathrm{BC}}{\mathrm{BD}}$

3. Fractions Subtraction with Equal Denominators

$\dfrac{A}{B}-\dfrac{C}{B}=\dfrac{A-C}{B}$

4. Fractions Subtraction with Different Denominators

$\dfrac{A}{B}-\dfrac{C}{D}=\dfrac{A D-B C}{B D}$

Algebraic Fractions Examples With Answers

Q 1. Solve the expression: $\dfrac{2}{x}-\dfrac{5}{x+2}$

Ans: $=\dfrac{2(x+2)}{x(x+2)}-\dfrac{5 \times x}{x(x+2)}$

Multiply the brackets out on the numerator, but not the denominator as we are going to be subtracting the two numerators together so need to collect like terms and then factorise if required.

$=\dfrac{2 x+4}{x(x+2)}-\dfrac{5 x}{x(x+2)}$

$=\dfrac{2 x+4-5 x}{x(x+2)}$

$=\dfrac{4-3 x}{x(x+2)}$

Q 2. Solve the expression: $\dfrac{x}{y}+\dfrac{y}{x}$

Ans: $\dfrac{x}{y}+\dfrac{y}{x}$

$=\dfrac{x^2}{x y}+\dfrac{y^2}{x y}$

$=\dfrac{x^2+y^2}{x y}$

Q 3. Solve the expression: $\dfrac{p}{14} \times \dfrac{6}{p}$

Ans: $\dfrac{p}{14} \times \dfrac{6}{p}$

Here, p is in the numerator and denominator, so it will be cancelled out.

$=\dfrac{3}{7}$

Q 4. Solve the expression: $\dfrac{5(x-6)}{x+5} \div \dfrac{x-6}{12}$

Ans: $\dfrac{5(x-6)}{x+5} \div \dfrac{x-6}{12}$

Invert the second fraction and change $\div$ to $x$

$=\dfrac{5(x-6)}{x+5} \times \dfrac{12}{x-6}$

$=\dfrac{60}{x+5}$

Practise Algebraic Fractions Worksheet

Q 1. Solve the expression: $\dfrac{x}{y}+\dfrac{y}{x}$

Ans: $\dfrac{x^2+y^2}{x y}$

Q 2. Solve the expression: $\dfrac{x-3}{4} \div \dfrac{x}{8}$

Ans: $\dfrac{2(x-3)}{x}$

Q 3. Solve the expression: $\dfrac{a}{7} \times \dfrac{b}{8}$

Ans: $\dfrac{a b}{56}$

Summary

Fractions in Algebra are used for probability estimation. They are used to indicate a numerical significance to the chance of the occurrence of an event. The denominator of an algebraic fraction is referred to as the denominator, while the dividend is referred to as the numerator. The terms of the algebraic fraction are the numerator and denominator. After clearing your concept, you can easily solve any adding and subtracting algebraic fractions worksheet. Algebraic fractions are subject to the same laws as arithmetic fractions. Operations of algebraic fractions includes simplifying algebraic fractions, adding and subtracting algebraic fractions and dividing algebraic fractions.